Discrete Morse Theory and Generalized Factor Order Bruce Sagan - PowerPoint PPT Presentation

Let P be a poset. The M¨ obius function of [ x , y ] ⊆ P is � µ ( x , y ) = − µ ( x , x ) = 1 , µ ( x , z ) . x ≤ z < y The order complex of [ x , y ] is the abstract simplicial complex ∆( x , y ) = { C : C is a chain in ( x , y ) } . Ex. The µ ( x , w ) in the following interval are purple. y 0 d [ x , y ] = 0 d c ∆( x , y ) = a 1 b a − 1 − 1 b c µ ( x , y ) = 0 x 1

Let P be a poset. The M¨ obius function of [ x , y ] ⊆ P is � µ ( x , y ) = − µ ( x , x ) = 1 , µ ( x , z ) . x ≤ z < y The order complex of [ x , y ] is the abstract simplicial complex ∆( x , y ) = { C : C is a chain in ( x , y ) } . Theorem µ ( x , y ) = ˜ χ (∆( x , y )) . Ex. The µ ( x , w ) in the following interval are purple. y 0 d [ x , y ] = 0 d c ∆( x , y ) = a 1 b a − 1 − 1 b c µ ( x , y ) = 0 x 1

Let P be a poset. The M¨ obius function of [ x , y ] ⊆ P is � µ ( x , y ) = − µ ( x , x ) = 1 , µ ( x , z ) . x ≤ z < y The order complex of [ x , y ] is the abstract simplicial complex ∆( x , y ) = { C : C is a chain in ( x , y ) } . Theorem µ ( x , y ) = ˜ χ (∆( x , y )) . Ex. The µ ( x , w ) in the following interval are purple. y 0 d [ x , y ] = 0 d c ∆( x , y ) = a 1 b a − 1 − 1 b c µ ( x , y ) = 0 χ (∆( x , y )) = 0 ˜ x 1

Outline Introduction to Forman’s Discrete Morse Theory (DMT) The M¨ obius function and the order complex ∆( x , y ) Babson and Hersh apply DMT to ∆( x , y ) Generalized Factor Order References

Let C = C ( x , y ) be the set of containment-maximal chains in ( x , y ) .

Let C = C ( x , y ) be the set of containment-maximal chains in ( x , y ) . By convention, list the elements of C ∈ C from smallest to largest including x , y .

Let C = C ( x , y ) be the set of containment-maximal chains in ( x , y ) . By convention, list the elements of C ∈ C from smallest to largest including x , y . If � is a total order on C then a new face of C ∈ C is C ′ ⊆ C with C ′ �⊆ B for all B ≺ C .

Let C = C ( x , y ) be the set of containment-maximal chains in ( x , y ) . By convention, list the elements of C ∈ C from smallest to largest including x , y . If � is a total order on C then a new face of C ∈ C is C ′ ⊆ C with C ′ �⊆ B for all B ≺ C . Ex. If B : x , a , c , d , y and C : x , b , c , d , y then the new faces of C are those containing b .

Let C = C ( x , y ) be the set of containment-maximal chains in ( x , y ) . By convention, list the elements of C ∈ C from smallest to largest including x , y . If � is a total order on C then a new face of C ∈ C is C ′ ⊆ C with C ′ �⊆ B for all B ≺ C . Ex. If B : x , a , c , d , y and C : x , b , c , d , y then the new faces of C are those containing b . We wish to construct a MM inductively by matching all new faces of each C except perhaps one. (1)

Let C = C ( x , y ) be the set of containment-maximal chains in ( x , y ) . By convention, list the elements of C ∈ C from smallest to largest including x , y . If � is a total order on C then a new face of C ∈ C is C ′ ⊆ C with C ′ �⊆ B for all B ≺ C . Ex. If B : x , a , c , d , y and C : x , b , c , d , y then the new faces of C are those containing b . We wish to construct a MM inductively by matching all new faces of each C except perhaps one. (1) B : x = x 0 , x 1 , . . . , x m = y and C : x = y 0 , y 1 , . . . , y n = y agree to level k if x 0 = y 0 , . . . , x k = y k .

Let C = C ( x , y ) be the set of containment-maximal chains in ( x , y ) . By convention, list the elements of C ∈ C from smallest to largest including x , y . If � is a total order on C then a new face of C ∈ C is C ′ ⊆ C with C ′ �⊆ B for all B ≺ C . Ex. If B : x , a , c , d , y and C : x , b , c , d , y then the new faces of C are those containing b . We wish to construct a MM inductively by matching all new faces of each C except perhaps one. (1) B : x = x 0 , x 1 , . . . , x m = y and C : x = y 0 , y 1 , . . . , y n = y agree to level k if x 0 = y 0 , . . . , x k = y k . They diverge from level k if they agree to level k but x k + 1 � = y k + 1 .

Let C = C ( x , y ) be the set of containment-maximal chains in ( x , y ) . By convention, list the elements of C ∈ C from smallest to largest including x , y . If � is a total order on C then a new face of C ∈ C is C ′ ⊆ C with C ′ �⊆ B for all B ≺ C . Ex. If B : x , a , c , d , y and C : x , b , c , d , y then the new faces of C are those containing b . We wish to construct a MM inductively by matching all new faces of each C except perhaps one. (1) B : x = x 0 , x 1 , . . . , x m = y and C : x = y 0 , y 1 , . . . , y n = y agree to level k if x 0 = y 0 , . . . , x k = y k . They diverge from level k if they agree to level k but x k + 1 � = y k + 1 . Ex. B : x , a , c , d , y and C : x , b , c , d , y diverge from level 0.

Let C = C ( x , y ) be the set of containment-maximal chains in ( x , y ) . By convention, list the elements of C ∈ C from smallest to largest including x , y . If � is a total order on C then a new face of C ∈ C is C ′ ⊆ C with C ′ �⊆ B for all B ≺ C . Ex. If B : x , a , c , d , y and C : x , b , c , d , y then the new faces of C are those containing b . We wish to construct a MM inductively by matching all new faces of each C except perhaps one. (1) B : x = x 0 , x 1 , . . . , x m = y and C : x = y 0 , y 1 , . . . , y n = y agree to level k if x 0 = y 0 , . . . , x k = y k . They diverge from level k if they agree to level k but x k + 1 � = y k + 1 . Ex. B : x , a , c , d , y and C : x , b , c , d , y diverge from level 0. Call � a poset lexicographic (PL) order if, whenever C , D diverge from some level k and C ′ , D ′ agree with C , D respectively to level k + 1, then ⇒ C ′ � D ′ . C � D ⇐

Let C = C ( x , y ) be the set of containment-maximal chains in ( x , y ) . By convention, list the elements of C ∈ C from smallest to largest including x , y . If � is a total order on C then a new face of C ∈ C is C ′ ⊆ C with C ′ �⊆ B for all B ≺ C . Ex. If B : x , a , c , d , y and C : x , b , c , d , y then the new faces of C are those containing b . We wish to construct a MM inductively by matching all new faces of each C except perhaps one. (1) B : x = x 0 , x 1 , . . . , x m = y and C : x = y 0 , y 1 , . . . , y n = y agree to level k if x 0 = y 0 , . . . , x k = y k . They diverge from level k if they agree to level k but x k + 1 � = y k + 1 . Ex. B : x , a , c , d , y and C : x , b , c , d , y diverge from level 0. Call � a poset lexicographic (PL) order if, whenever C , D diverge from some level k and C ′ , D ′ agree with C , D respectively to level k + 1, then ⇒ C ′ � D ′ . C � D ⇐ Proposition (Babson and Hersh) If � is an PL-order then it has a MM satisfying (1).

How do we identify new faces?

How do we identify new faces? A closed interval in a chain C : x 0 , . . . , x n is a subchain of the form I = C [ x k , x l ] : x k , x k + 1 , . . . , x l .

How do we identify new faces? A closed interval in a chain C : x 0 , . . . , x n is a subchain of the form I = C [ x k , x l ] : x k , x k + 1 , . . . , x l . Open intervals C ( x k , x l ) are defined similarly.

How do we identify new faces? A closed interval in a chain C : x 0 , . . . , x n is a subchain of the form I = C [ x k , x l ] : x k , x k + 1 , . . . , x l . Open intervals C ( x k , x l ) are defined similarly. A skipped interval (SI) is I ⊆ C with C − I ⊆ B for some B ≺ C .

How do we identify new faces? A closed interval in a chain C : x 0 , . . . , x n is a subchain of the form I = C [ x k , x l ] : x k , x k + 1 , . . . , x l . Open intervals C ( x k , x l ) are defined similarly. A skipped interval (SI) is I ⊆ C with C − I ⊆ B for some B ≺ C . A minimal skipped interval (MSI) is a SI which is minimal with respect to containment.

How do we identify new faces? A closed interval in a chain C : x 0 , . . . , x n is a subchain of the form I = C [ x k , x l ] : x k , x k + 1 , . . . , x l . Open intervals C ( x k , x l ) are defined similarly. A skipped interval (SI) is I ⊆ C with C − I ⊆ B for some B ≺ C . A minimal skipped interval (MSI) is a SI which is minimal with respect to containment. Ex. If B : x , a , c , d , y & C : x , b , c , d , y then C has MSI { b } .

How do we identify new faces? A closed interval in a chain C : x 0 , . . . , x n is a subchain of the form I = C [ x k , x l ] : x k , x k + 1 , . . . , x l . Open intervals C ( x k , x l ) are defined similarly. A skipped interval (SI) is I ⊆ C with C − I ⊆ B for some B ≺ C . A minimal skipped interval (MSI) is a SI which is minimal with respect to containment. Ex. If B : x , a , c , d , y & C : x , b , c , d , y then C has MSI { b } . I ( C ) def = { I : I is an MSI of C } .

How do we identify new faces? A closed interval in a chain C : x 0 , . . . , x n is a subchain of the form I = C [ x k , x l ] : x k , x k + 1 , . . . , x l . Open intervals C ( x k , x l ) are defined similarly. A skipped interval (SI) is I ⊆ C with C − I ⊆ B for some B ≺ C . A minimal skipped interval (MSI) is a SI which is minimal with respect to containment. Ex. If B : x , a , c , d , y & C : x , b , c , d , y then C has MSI { b } . I ( C ) def = { I : I is an MSI of C } . Lemma (Babson and Hersh) C ′ ⊆ C is new ⇐ ⇒ C ′ ∩ I � = ∅ for all I ∈ I ( C ) .

How do we identify new faces? A closed interval in a chain C : x 0 , . . . , x n is a subchain of the form I = C [ x k , x l ] : x k , x k + 1 , . . . , x l . Open intervals C ( x k , x l ) are defined similarly. A skipped interval (SI) is I ⊆ C with C − I ⊆ B for some B ≺ C . A minimal skipped interval (MSI) is a SI which is minimal with respect to containment. Ex. If B : x , a , c , d , y & C : x , b , c , d , y then C has MSI { b } . I ( C ) def = { I : I is an MSI of C } . Lemma (Babson and Hersh) C ′ ⊆ C is new ⇐ ⇒ C ′ ∩ I � = ∅ for all I ∈ I ( C ) . Proof “ = ⇒ ”

How do we identify new faces? A closed interval in a chain C : x 0 , . . . , x n is a subchain of the form I = C [ x k , x l ] : x k , x k + 1 , . . . , x l . Open intervals C ( x k , x l ) are defined similarly. A skipped interval (SI) is I ⊆ C with C − I ⊆ B for some B ≺ C . A minimal skipped interval (MSI) is a SI which is minimal with respect to containment. Ex. If B : x , a , c , d , y & C : x , b , c , d , y then C has MSI { b } . I ( C ) def = { I : I is an MSI of C } . Lemma (Babson and Hersh) C ′ ⊆ C is new ⇐ ⇒ C ′ ∩ I � = ∅ for all I ∈ I ( C ) . ⇒ ” If C ′ ∩ I = ∅ for some I then C ′ ⊆ C − I . Proof “ =

How do we identify new faces? A closed interval in a chain C : x 0 , . . . , x n is a subchain of the form I = C [ x k , x l ] : x k , x k + 1 , . . . , x l . Open intervals C ( x k , x l ) are defined similarly. A skipped interval (SI) is I ⊆ C with C − I ⊆ B for some B ≺ C . A minimal skipped interval (MSI) is a SI which is minimal with respect to containment. Ex. If B : x , a , c , d , y & C : x , b , c , d , y then C has MSI { b } . I ( C ) def = { I : I is an MSI of C } . Lemma (Babson and Hersh) C ′ ⊆ C is new ⇐ ⇒ C ′ ∩ I � = ∅ for all I ∈ I ( C ) . ⇒ ” If C ′ ∩ I = ∅ for some I then C ′ ⊆ C − I . And I a Proof “ = skipped interval implies C − I ⊆ B for some B ≺ C .

How do we identify new faces? A closed interval in a chain C : x 0 , . . . , x n is a subchain of the form I = C [ x k , x l ] : x k , x k + 1 , . . . , x l . Open intervals C ( x k , x l ) are defined similarly. A skipped interval (SI) is I ⊆ C with C − I ⊆ B for some B ≺ C . A minimal skipped interval (MSI) is a SI which is minimal with respect to containment. Ex. If B : x , a , c , d , y & C : x , b , c , d , y then C has MSI { b } . I ( C ) def = { I : I is an MSI of C } . Lemma (Babson and Hersh) C ′ ⊆ C is new ⇐ ⇒ C ′ ∩ I � = ∅ for all I ∈ I ( C ) . ⇒ ” If C ′ ∩ I = ∅ for some I then C ′ ⊆ C − I . And I a Proof “ = skipped interval implies C − I ⊆ B for some B ≺ C . But then C ′ ⊆ B for B ≺ C , contradicting the fact that C ′ is new.

Call a C containing an unmatched face critical . How do we identify critical chains?

Call a C containing an unmatched face critical . How do we identify critical chains? We need to turn I ( C ) into a set of disjoint intervals J ( C ) as follows.

Call a C containing an unmatched face critical . How do we identify critical chains? We need to turn I ( C ) into a set of disjoint intervals J ( C ) as follows. Since I ( C ) has no containments, the intervals can be ordered I 1 , . . . , I l so that min I 1 < . . . < min I l and max I 1 < . . . < max I l .

Call a C containing an unmatched face critical . How do we identify critical chains? We need to turn I ( C ) into a set of disjoint intervals J ( C ) as follows. Since I ( C ) has no containments, the intervals can be ordered I 1 , . . . , I l so that min I 1 < . . . < min I l and max I 1 < . . . < max I l . Let J 1 = I 1 .

Call a C containing an unmatched face critical . How do we identify critical chains? We need to turn I ( C ) into a set of disjoint intervals J ( C ) as follows. Since I ( C ) has no containments, the intervals can be ordered I 1 , . . . , I l so that min I 1 < . . . < min I l and max I 1 < . . . < max I l . Let J 1 = I 1 . Construct I ′ 2 = I 2 − J 1 , . . . , I ′ l = I l − J 1 and throw out any which are not containment minimal.

Call a C containing an unmatched face critical . How do we identify critical chains? We need to turn I ( C ) into a set of disjoint intervals J ( C ) as follows. Since I ( C ) has no containments, the intervals can be ordered I 1 , . . . , I l so that min I 1 < . . . < min I l and max I 1 < . . . < max I l . Let J 1 = I 1 . Construct I ′ 2 = I 2 − J 1 , . . . , I ′ l = I l − J 1 and throw out any which are not containment minimal. Let J 2 = I ′ j where j is the smallest index of the intervals remaining.

Call a C containing an unmatched face critical . How do we identify critical chains? We need to turn I ( C ) into a set of disjoint intervals J ( C ) as follows. Since I ( C ) has no containments, the intervals can be ordered I 1 , . . . , I l so that min I 1 < . . . < min I l and max I 1 < . . . < max I l . Let J 1 = I 1 . Construct I ′ 2 = I 2 − J 1 , . . . , I ′ l = I l − J 1 and throw out any which are not containment minimal. Let J 2 = I ′ j where j is the smallest index of the intervals remaining. Continue in this way to form J ( C ) .

Call a C containing an unmatched face critical . How do we identify critical chains? We need to turn I ( C ) into a set of disjoint intervals J ( C ) as follows. Since I ( C ) has no containments, the intervals can be ordered I 1 , . . . , I l so that min I 1 < . . . < min I l and max I 1 < . . . < max I l . Let J 1 = I 1 . Construct I ′ 2 = I 2 − J 1 , . . . , I ′ l = I l − J 1 and throw out any which are not containment minimal. Let J 2 = I ′ j where j is the smallest index of the intervals remaining. Continue in this way to form J ( C ) . Theorem (Babson and Hersh) Let [ x , y ] be an interval and let ≺ be an PL order on C ( x , y ) .

Call a C containing an unmatched face critical . How do we identify critical chains? We need to turn I ( C ) into a set of disjoint intervals J ( C ) as follows. Since I ( C ) has no containments, the intervals can be ordered I 1 , . . . , I l so that min I 1 < . . . < min I l and max I 1 < . . . < max I l . Let J 1 = I 1 . Construct I ′ 2 = I 2 − J 1 , . . . , I ′ l = I l − J 1 and throw out any which are not containment minimal. Let J 2 = I ′ j where j is the smallest index of the intervals remaining. Continue in this way to form J ( C ) . Theorem (Babson and Hersh) Let [ x , y ] be an interval and let ≺ be an PL order on C ( x , y ) . 1. C ∈ C ( x , y ) is critical ⇐ ⇒ J ( C ) covers C.

Call a C containing an unmatched face critical . How do we identify critical chains? We need to turn I ( C ) into a set of disjoint intervals J ( C ) as follows. Since I ( C ) has no containments, the intervals can be ordered I 1 , . . . , I l so that min I 1 < . . . < min I l and max I 1 < . . . < max I l . Let J 1 = I 1 . Construct I ′ 2 = I 2 − J 1 , . . . , I ′ l = I l − J 1 and throw out any which are not containment minimal. Let J 2 = I ′ j where j is the smallest index of the intervals remaining. Continue in this way to form J ( C ) . Theorem (Babson and Hersh) Let [ x , y ] be an interval and let ≺ be an PL order on C ( x , y ) . 1. C ∈ C ( x , y ) is critical ⇐ ⇒ J ( C ) covers C. 2. The critical face of a critical chain C is obtained by picking the smallest element from each J ∈ J ( C ) .

Call a C containing an unmatched face critical . How do we identify critical chains? We need to turn I ( C ) into a set of disjoint intervals J ( C ) as follows. Since I ( C ) has no containments, the intervals can be ordered I 1 , . . . , I l so that min I 1 < . . . < min I l and max I 1 < . . . < max I l . Let J 1 = I 1 . Construct I ′ 2 = I 2 − J 1 , . . . , I ′ l = I l − J 1 and throw out any which are not containment minimal. Let J 2 = I ′ j where j is the smallest index of the intervals remaining. Continue in this way to form J ( C ) . Theorem (Babson and Hersh) Let [ x , y ] be an interval and let ≺ be an PL order on C ( x , y ) . 1. C ∈ C ( x , y ) is critical ⇐ ⇒ J ( C ) covers C. 2. The critical face of a critical chain C is obtained by picking the smallest element from each J ∈ J ( C ) . 3. We have � ( − 1 ) # J ( C ) − 1 µ ( x , y ) = C where the sum is over all critical C ∈ C ( x , y ) .

Outline Introduction to Forman’s Discrete Morse Theory (DMT) The M¨ obius function and the order complex ∆( x , y ) Babson and Hersh apply DMT to ∆( x , y ) Generalized Factor Order References

Let A be a set (the alphabet ) and let A ∗ be the set of words w over A .

Let A be a set (the alphabet ) and let A ∗ be the set of words w over A . Call u ∈ A ∗ a factor of w if w = xuy for some x , y ∈ A ∗ .

Let A be a set (the alphabet ) and let A ∗ be the set of words w over A . Call u ∈ A ∗ a factor of w if w = xuy for some x , y ∈ A ∗ . Ex. u = abba is a factor of w = baabbaa .

Let A be a set (the alphabet ) and let A ∗ be the set of words w over A . Call u ∈ A ∗ a factor of w if w = xuy for some x , y ∈ A ∗ . Ex. u = abba is a factor of w = baabbaa . Factor order on A ∗ is the partial order u ≤ w if u is a factor of w .

Let A be a set (the alphabet ) and let A ∗ be the set of words w over A . Call u ∈ A ∗ a factor of w if w = xuy for some x , y ∈ A ∗ . Ex. u = abba is a factor of w = baabbaa . Factor order on A ∗ is the partial order u ≤ w if u is a factor of w . The inner and outer factors of w = a 1 a 2 . . . a n are i ( w ) = a 2 . . . a n − 1 .

Let A be a set (the alphabet ) and let A ∗ be the set of words w over A . Call u ∈ A ∗ a factor of w if w = xuy for some x , y ∈ A ∗ . Ex. u = abba is a factor of w = baabbaa . Factor order on A ∗ is the partial order u ≤ w if u is a factor of w . The inner and outer factors of w = a 1 a 2 . . . a n are i ( w ) = a 2 . . . a n − 1 . o ( w ) = longest word which is a proper prefix and suffix of w .

Let A be a set (the alphabet ) and let A ∗ be the set of words w over A . Call u ∈ A ∗ a factor of w if w = xuy for some x , y ∈ A ∗ . Ex. u = abba is a factor of w = baabbaa . Factor order on A ∗ is the partial order u ≤ w if u is a factor of w . The inner and outer factors of w = a 1 a 2 . . . a n are i ( w ) = a 2 . . . a n − 1 . o ( w ) = longest word which is a proper prefix and suffix of w . Ex. w = abbab has i ( w ) = bba and o ( w ) = ab .

Let A be a set (the alphabet ) and let A ∗ be the set of words w over A . Call u ∈ A ∗ a factor of w if w = xuy for some x , y ∈ A ∗ . Ex. u = abba is a factor of w = baabbaa . Factor order on A ∗ is the partial order u ≤ w if u is a factor of w . The inner and outer factors of w = a 1 a 2 . . . a n are i ( w ) = a 2 . . . a n − 1 . o ( w ) = longest word which is a proper prefix and suffix of w . Ex. w = abbab has i ( w ) = bba and o ( w ) = ab . Call w = a 1 . . . a n flat if a 1 = . . . = a n .

Let A be a set (the alphabet ) and let A ∗ be the set of words w over A . Call u ∈ A ∗ a factor of w if w = xuy for some x , y ∈ A ∗ . Ex. u = abba is a factor of w = baabbaa . Factor order on A ∗ is the partial order u ≤ w if u is a factor of w . The inner and outer factors of w = a 1 a 2 . . . a n are i ( w ) = a 2 . . . a n − 1 . o ( w ) = longest word which is a proper prefix and suffix of w . Ex. w = abbab has i ( w ) = bba and o ( w ) = ab . Call w = a 1 . . . a n flat if a 1 = . . . = a n . Let | w | be w ’s length.

Let A be a set (the alphabet ) and let A ∗ be the set of words w over A . Call u ∈ A ∗ a factor of w if w = xuy for some x , y ∈ A ∗ . Ex. u = abba is a factor of w = baabbaa . Factor order on A ∗ is the partial order u ≤ w if u is a factor of w . The inner and outer factors of w = a 1 a 2 . . . a n are i ( w ) = a 2 . . . a n − 1 . o ( w ) = longest word which is a proper prefix and suffix of w . Ex. w = abbab has i ( w ) = bba and o ( w ) = ab . Call w = a 1 . . . a n flat if a 1 = . . . = a n . Let | w | be w ’s length. Theorem (Bj¨ orner) In factor order on A ∗  µ ( u , o ( w )) if | w | − | u | > 2 , u ≤ o ( w ) �≤ i ( w ) ;     if | w | − | u | = 2 , w not flat, u ∈ { o ( w ) , i ( w ) } ; 1  µ ( u , w ) = ( − 1 ) | w |−| u | if | w | − | u | < 2 ;     0 otherwise. 

Let A be a set (the alphabet ) and let A ∗ be the set of words w over A . Call u ∈ A ∗ a factor of w if w = xuy for some x , y ∈ A ∗ . Ex. u = abba is a factor of w = baabbaa . Factor order on A ∗ is the partial order u ≤ w if u is a factor of w . The inner and outer factors of w = a 1 a 2 . . . a n are i ( w ) = a 2 . . . a n − 1 . o ( w ) = longest word which is a proper prefix and suffix of w . Ex. w = abbab has i ( w ) = bba and o ( w ) = ab . Call w = a 1 . . . a n flat if a 1 = . . . = a n . Let | w | be w ’s length. Theorem (Bj¨ orner) In factor order on A ∗  µ ( u , o ( w )) if | w | − | u | > 2 , u ≤ o ( w ) �≤ i ( w ) ; ⇐     if | w | − | u | = 2 , w not flat, u ∈ { o ( w ) , i ( w ) } ; 1  µ ( u , w ) = ( − 1 ) | w |−| u | if | w | − | u | < 2 ;     0 otherwise. ⇐ 

Let A be a set (the alphabet ) and let A ∗ be the set of words w over A . Call u ∈ A ∗ a factor of w if w = xuy for some x , y ∈ A ∗ . Ex. u = abba is a factor of w = baabbaa . Factor order on A ∗ is the partial order u ≤ w if u is a factor of w . The inner and outer factors of w = a 1 a 2 . . . a n are i ( w ) = a 2 . . . a n − 1 . o ( w ) = longest word which is a proper prefix and suffix of w . Ex. w = abbab has i ( w ) = bba and o ( w ) = ab . Call w = a 1 . . . a n flat if a 1 = . . . = a n . Let | w | be w ’s length. Theorem (Bj¨ orner) In factor order on A ∗  µ ( u , o ( w )) if | w | − | u | > 2 , u ≤ o ( w ) �≤ i ( w ) ; ⇐     if | w | − | u | = 2 , w not flat, u ∈ { o ( w ) , i ( w ) } ; 1  µ ( u , w ) = ( − 1 ) | w |−| u | if | w | − | u | < 2 ;     0 otherwise. ⇐  Also, ∆( u , w ) ≃ ball or sphere when µ ( u , w ) = 0 or ± 1 , resp.

Let A be a set (the alphabet ) and let A ∗ be the set of words w over A . Call u ∈ A ∗ a factor of w if w = xuy for some x , y ∈ A ∗ . Ex. u = abba is a factor of w = baabbaa . Factor order on A ∗ is the partial order u ≤ w if u is a factor of w . The inner and outer factors of w = a 1 a 2 . . . a n are i ( w ) = a 2 . . . a n − 1 . o ( w ) = longest word which is a proper prefix and suffix of w . Ex. w = abbab has i ( w ) = bba and o ( w ) = ab . Call w = a 1 . . . a n flat if a 1 = . . . = a n . Let | w | be w ’s length. Theorem (Bj¨ orner) In factor order on A ∗  µ ( u , o ( w )) if | w | − | u | > 2 , u ≤ o ( w ) �≤ i ( w ) ; ⇐     if | w | − | u | = 2 , w not flat, u ∈ { o ( w ) , i ( w ) } ; 1  µ ( u , w ) = ( − 1 ) | w |−| u | if | w | − | u | < 2 ;     0 otherwise. ⇐  Also, ∆( u , w ) ≃ ball or sphere when µ ( u , w ) = 0 or ± 1 , resp. Ex. µ ( a , abbab )

Let A be a set (the alphabet ) and let A ∗ be the set of words w over A . Call u ∈ A ∗ a factor of w if w = xuy for some x , y ∈ A ∗ . Ex. u = abba is a factor of w = baabbaa . Factor order on A ∗ is the partial order u ≤ w if u is a factor of w . The inner and outer factors of w = a 1 a 2 . . . a n are i ( w ) = a 2 . . . a n − 1 . o ( w ) = longest word which is a proper prefix and suffix of w . Ex. w = abbab has i ( w ) = bba and o ( w ) = ab . Call w = a 1 . . . a n flat if a 1 = . . . = a n . Let | w | be w ’s length. Theorem (Bj¨ orner) In factor order on A ∗  µ ( u , o ( w )) if | w | − | u | > 2 , u ≤ o ( w ) �≤ i ( w ) ; ⇐     if | w | − | u | = 2 , w not flat, u ∈ { o ( w ) , i ( w ) } ; 1  µ ( u , w ) = ( − 1 ) | w |−| u | if | w | − | u | < 2 ;     0 otherwise. ⇐  Also, ∆( u , w ) ≃ ball or sphere when µ ( u , w ) = 0 or ± 1 , resp. Ex. µ ( a , abbab ) = µ ( a , ab )

Let A be a set (the alphabet ) and let A ∗ be the set of words w over A . Call u ∈ A ∗ a factor of w if w = xuy for some x , y ∈ A ∗ . Ex. u = abba is a factor of w = baabbaa . Factor order on A ∗ is the partial order u ≤ w if u is a factor of w . The inner and outer factors of w = a 1 a 2 . . . a n are i ( w ) = a 2 . . . a n − 1 . o ( w ) = longest word which is a proper prefix and suffix of w . Ex. w = abbab has i ( w ) = bba and o ( w ) = ab . Call w = a 1 . . . a n flat if a 1 = . . . = a n . Let | w | be w ’s length. Theorem (Bj¨ orner) In factor order on A ∗  µ ( u , o ( w )) if | w | − | u | > 2 , u ≤ o ( w ) �≤ i ( w ) ; ⇐     if | w | − | u | = 2 , w not flat, u ∈ { o ( w ) , i ( w ) } ; 1  µ ( u , w ) = ( − 1 ) | w |−| u | if | w | − | u | < 2 ;     0 otherwise. ⇐  Also, ∆( u , w ) ≃ ball or sphere when µ ( u , w ) = 0 or ± 1 , resp. Ex. µ ( a , abbab ) = µ ( a , ab ) = − 1.

Write chains in [ u , w ] dually from largest to smallest element.

Write chains in [ u , w ] dually from largest to smallest element. An embedding of u in w is η ∈ ( A ⊎ { 0 } ) ∗ obtained by zeroing out the positions of w outside of a given factor equal to u .

Write chains in [ u , w ] dually from largest to smallest element. An embedding of u in w is η ∈ ( A ⊎ { 0 } ) ∗ obtained by zeroing out the positions of w outside of a given factor equal to u . Ex. If u = abba and w = baabbaa then η = 00 abba 0

Write chains in [ u , w ] dually from largest to smallest element. An embedding of u in w is η ∈ ( A ⊎ { 0 } ) ∗ obtained by zeroing out the positions of w outside of a given factor equal to u . Ex. If u = abba and w = baabbaa then η = 00 abba 0 If y covers x then there is a unique embedding of x in y , unless y is flat in which case we choose the embedding starting with 0.

Write chains in [ u , w ] dually from largest to smallest element. An embedding of u in w is η ∈ ( A ⊎ { 0 } ) ∗ obtained by zeroing out the positions of w outside of a given factor equal to u . Ex. If u = abba and w = baabbaa then η = 00 abba 0 If y covers x then there is a unique embedding of x in y , unless y is flat in which case we choose the embedding starting with 0. So any maximal chain C : w = w 0 , w 1 , . . . , w m = u determines a chain of embeddings with labels l ( C ) = ( l 1 , . . . , l n ) l 1 l 2 l 3 → . . . l m → η 1 → η 2 → η m C : η 0 where the l i give the position of the new zero in η i .

Write chains in [ u , w ] dually from largest to smallest element. An embedding of u in w is η ∈ ( A ⊎ { 0 } ) ∗ obtained by zeroing out the positions of w outside of a given factor equal to u . Ex. If u = abba and w = baabbaa then η = 00 abba 0 If y covers x then there is a unique embedding of x in y , unless y is flat in which case we choose the embedding starting with 0. So any maximal chain C : w = w 0 , w 1 , . . . , w m = u determines a chain of embeddings with labels l ( C ) = ( l 1 , . . . , l n ) l 1 l 2 l 3 → . . . l m → η 1 → η 2 → η m C : η 0 where the l i give the position of the new zero in η i . Ex. C : baabbaa , aabbaa , aabba , abba becomes C : baabbaa 1 → 0 aabbaa 7 → 0 aabba 0 2 → 00 abba 0,

Write chains in [ u , w ] dually from largest to smallest element. An embedding of u in w is η ∈ ( A ⊎ { 0 } ) ∗ obtained by zeroing out the positions of w outside of a given factor equal to u . Ex. If u = abba and w = baabbaa then η = 00 abba 0 If y covers x then there is a unique embedding of x in y , unless y is flat in which case we choose the embedding starting with 0. So any maximal chain C : w = w 0 , w 1 , . . . , w m = u determines a chain of embeddings with labels l ( C ) = ( l 1 , . . . , l n ) l 1 l 2 l 3 → . . . l m → η 1 → η 2 → η m C : η 0 where the l i give the position of the new zero in η i . Ex. C : baabbaa , aabbaa , aabba , abba becomes C : baabbaa 1 → 0 aabbaa 7 → 0 aabba 0 2 → 00 abba 0, l ( C ) = ( 1 , 7 , 2 ) .

Write chains in [ u , w ] dually from largest to smallest element. An embedding of u in w is η ∈ ( A ⊎ { 0 } ) ∗ obtained by zeroing out the positions of w outside of a given factor equal to u . Ex. If u = abba and w = baabbaa then η = 00 abba 0 If y covers x then there is a unique embedding of x in y , unless y is flat in which case we choose the embedding starting with 0. So any maximal chain C : w = w 0 , w 1 , . . . , w m = u determines a chain of embeddings with labels l ( C ) = ( l 1 , . . . , l n ) l 1 l 2 l 3 → . . . l m → η 1 → η 2 → η m C : η 0 where the l i give the position of the new zero in η i . Ex. C : baabbaa , aabbaa , aabba , abba becomes C : baabbaa 1 → 0 aabbaa 7 → 0 aabba 0 2 → 00 abba 0, l ( C ) = ( 1 , 7 , 2 ) . Lemma (S and Willenbring) The total order on C ( w , u ) given by B � C iff l ( B ) ≤ lex l ( C ) is a PL-order

DMT gives a proof of Bj¨ orner’s formula which explains the definitions of i ( w ) and o ( w ) and the inequality between them.

DMT gives a proof of Bj¨ orner’s formula which explains the definitions of i ( w ) and o ( w ) and the inequality between them. Ex. Let u = a , w = abbab and consider all chains in C ( w , u ) passing through ab = o ( w ) : 1 2 3 5 B : abbab → 0 bbab → 00 bab → 000 ab → 000 a 0 , → abba 0 4 5 → abb 00 3 → ab 000 2 C : abbab → a 0000 .

DMT gives a proof of Bj¨ orner’s formula which explains the definitions of i ( w ) and o ( w ) and the inequality between them. Ex. Let u = a , w = abbab and consider all chains in C ( w , u ) passing through ab = o ( w ) : 1 2 3 5 B : abbab → 0 bbab → 00 bab → 000 ab → 000 a 0 , → abba 0 4 5 → abb 00 3 → ab 000 2 C : abbab → a 0000 . Note that C ( abbab , ab ) is an SI of C and is, in fact, an MSI.

DMT gives a proof of Bj¨ orner’s formula which explains the definitions of i ( w ) and o ( w ) and the inequality between them. Ex. Let u = a , w = abbab and consider all chains in C ( w , u ) passing through ab = o ( w ) : 1 2 3 5 B : abbab → 0 bbab → 00 bab → 000 ab → 000 a 0 , → abba 0 4 5 → abb 00 3 → ab 000 2 C : abbab → a 0000 . Note that C ( abbab , ab ) is an SI of C and is, in fact, an MSI. Proposition (S and Willenbring) Let u ≤ o ( w ) �≤ i ( w ) and let C ∈ C ( w , u ) be the lexicographically first chain passing through the prefix embedding of o ( w ) in w.

DMT gives a proof of Bj¨ orner’s formula which explains the definitions of i ( w ) and o ( w ) and the inequality between them. Ex. Let u = a , w = abbab and consider all chains in C ( w , u ) passing through ab = o ( w ) : 1 2 3 5 B : abbab → 0 bbab → 00 bab → 000 ab → 000 a 0 , → abba 0 4 5 → abb 00 3 → ab 000 2 C : abbab → a 0000 . Note that C ( abbab , ab ) is an SI of C and is, in fact, an MSI. Proposition (S and Willenbring) Let u ≤ o ( w ) �≤ i ( w ) and let C ∈ C ( w , u ) be the lexicographically first chain passing through the prefix embedding of o ( w ) in w. Then I = C ( w , o ( w )) is an MSI.

DMT gives a proof of Bj¨ orner’s formula which explains the definitions of i ( w ) and o ( w ) and the inequality between them. Ex. Let u = a , w = abbab and consider all chains in C ( w , u ) passing through ab = o ( w ) : 1 2 3 5 B : abbab → 0 bbab → 00 bab → 000 ab → 000 a 0 , → abba 0 4 5 → abb 00 3 → ab 000 2 C : abbab → a 0000 . Note that C ( abbab , ab ) is an SI of C and is, in fact, an MSI. Proposition (S and Willenbring) Let u ≤ o ( w ) �≤ i ( w ) and let C ∈ C ( w , u ) be the lexicographically first chain passing through the prefix embedding of o ( w ) in w. Then I = C ( w , o ( w )) is an MSI. Proof Let B be the chain which goes from w to the suffix embedding of o ( w ) and then continues to u as does C .

DMT gives a proof of Bj¨ orner’s formula which explains the definitions of i ( w ) and o ( w ) and the inequality between them. Ex. Let u = a , w = abbab and consider all chains in C ( w , u ) passing through ab = o ( w ) : 1 2 3 5 B : abbab → 0 bbab → 00 bab → 000 ab → 000 a 0 , → abba 0 4 5 → abb 00 3 → ab 000 2 C : abbab → a 0000 . Note that C ( abbab , ab ) is an SI of C and is, in fact, an MSI. Proposition (S and Willenbring) Let u ≤ o ( w ) �≤ i ( w ) and let C ∈ C ( w , u ) be the lexicographically first chain passing through the prefix embedding of o ( w ) in w. Then I = C ( w , o ( w )) is an MSI. Proof Let B be the chain which goes from w to the suffix embedding of o ( w ) and then continues to u as does C . Then B ≺ C and C − I ⊆ B so I is an SI.

DMT gives a proof of Bj¨ orner’s formula which explains the definitions of i ( w ) and o ( w ) and the inequality between them. Ex. Let u = a , w = abbab and consider all chains in C ( w , u ) passing through ab = o ( w ) : 1 2 3 5 B : abbab → 0 bbab → 00 bab → 000 ab → 000 a 0 , → abba 0 4 5 → abb 00 3 → ab 000 2 C : abbab → a 0000 . Note that C ( abbab , ab ) is an SI of C and is, in fact, an MSI. Proposition (S and Willenbring) Let u ≤ o ( w ) �≤ i ( w ) and let C ∈ C ( w , u ) be the lexicographically first chain passing through the prefix embedding of o ( w ) in w. Then I = C ( w , o ( w )) is an MSI. Proof Let B be the chain which goes from w to the suffix embedding of o ( w ) and then continues to u as does C . Then B ≺ C and C − I ⊆ B so I is an SI. Because o ( w ) �≤ i ( w ) there are only two embeddings of o ( w ) in w : prefix and suffix.

DMT gives a proof of Bj¨ orner’s formula which explains the definitions of i ( w ) and o ( w ) and the inequality between them. Ex. Let u = a , w = abbab and consider all chains in C ( w , u ) passing through ab = o ( w ) : 1 2 3 5 B : abbab → 0 bbab → 00 bab → 000 ab → 000 a 0 , → abba 0 4 5 → abb 00 3 → ab 000 2 C : abbab → a 0000 . Note that C ( abbab , ab ) is an SI of C and is, in fact, an MSI. Proposition (S and Willenbring) Let u ≤ o ( w ) �≤ i ( w ) and let C ∈ C ( w , u ) be the lexicographically first chain passing through the prefix embedding of o ( w ) in w. Then I = C ( w , o ( w )) is an MSI. Proof Let B be the chain which goes from w to the suffix embedding of o ( w ) and then continues to u as does C . Then B ≺ C and C − I ⊆ B so I is an SI. Because o ( w ) �≤ i ( w ) there are only two embeddings of o ( w ) in w : prefix and suffix. Thus, since C is lexicographically first through the prefix embedding, any other chain prior to C must agree with the portion of B up to the suffix embedding of o ( w ) .

DMT gives a proof of Bj¨ orner’s formula which explains the definitions of i ( w ) and o ( w ) and the inequality between them. Ex. Let u = a , w = abbab and consider all chains in C ( w , u ) passing through ab = o ( w ) : 1 2 3 5 B : abbab → 0 bbab → 00 bab → 000 ab → 000 a 0 , → abba 0 4 5 → abb 00 3 → ab 000 2 C : abbab → a 0000 . Note that C ( abbab , ab ) is an SI of C and is, in fact, an MSI. Proposition (S and Willenbring) Let u ≤ o ( w ) �≤ i ( w ) and let C ∈ C ( w , u ) be the lexicographically first chain passing through the prefix embedding of o ( w ) in w. Then I = C ( w , o ( w )) is an MSI. Proof Let B be the chain which goes from w to the suffix embedding of o ( w ) and then continues to u as does C . Then B ≺ C and C − I ⊆ B so I is an SI. Because o ( w ) �≤ i ( w ) there are only two embeddings of o ( w ) in w : prefix and suffix. Thus, since C is lexicographically first through the prefix embedding, any other chain prior to C must agree with the portion of B up to the suffix embedding of o ( w ) . So I is an MSI.